1. Field of the Invention
Embodiments of the present invention relate to digital transceivers for use in communication systems, and in preferred embodiments, to systems and methods for providing compensation for I-Q imbalance in digital transceivers.
2. Description of Related Art
In communication systems, quadrature up and down-conversion are needed to perform frequency translations. In general, the transmitter uses a quadrature up converter to convert a signal to a higher frequency while the receiver uses a quadrature down converter to convert a signal to a lower frequency. An example of a quadrature converter 102 is shown in FIG. 1. Quadrature converter 102 includes two mixers 104, 106 and an oscillator 108. The paths that are associated with the x and y inputs to mixers 104 and 106, respectively, are referred to as the I-channel and Q-channel. The oscillator 108 provides a cos ωot signal input to mixer 104 and a k sin ωot signal input to mixer 106, where k is an arbitrary constant. The output of the quadrature converter 102 may be viewed as a complex signal x cos ωot+jky sin ωot.
The quadrature converter may be used in multiple configurations. One common configuration is shown in FIG. 2, which shows quadrature down-converter 202. Quadrature down-converter 202 includes two mixers 204, 206 and an oscillator 208. For the configuration of the quadrature down-converter shown in FIG. 2, the inputs to mixers 204, 206 are x=y=r(t)cos(ωit), where ωi is the input carrier frequency. The oscillator 208 provides a cos wot signal input to mixer 204 and a k sin wot signal input to mixer 206, where k=−1. The output of quadrature down-converter 202 then becomes r(t)cos(ωit)e−jωot.
In the configuration shown in FIG. 2, the quadrature down-converter 202 may be viewed as a single sideband mixer because the quadrature oscillator behaves like a complex sinusoid e−jωot that has spectral content only at −ωo. A real sinusoid would have spectral content at ωo as well as at −ωo. The positive frequency component causes images to be folded in-band at the mixer output. In-band signal refers to the lower portion of the frequency content around DC (i.e. where the frequency equals zero Hz).
An image is defined as the spectral components located at 2ωo−ωi and −2ωo+ωi as shown in FIG. 3A, where the results of double sideband mixing are illustrated. In FIG. 3A, a real sinusoid cos ωot is used instead of e−jωot. FIG. 3A shows desired signals and images before mixing (top of FIG. 3A) and after mixing (bottom of FIG. 3A). It can be seen in FIG. 3A that, after mixing, the images are folded in-band at the mixer output.
FIG. 3B illustrates single sideband mixing and similarly shows desired signals and images before mixing (top of FIG. 3B) and after mixing (bottom of FIG. 3B). As can be seen in FIG. 3B, a single sideband mixer has the advantage that it reduces half of the spectral products and therefore eliminates the effect of image folding in-band. Folding of image in-band is a problem especially if the value of ωi−ωo is small. Such a situation arises in a low-IF super-heterodyne receiver. In a direct-conversion receiver, where ωi=ωo, an image does not exist. Thus, the single sideband mixer may function as a quadrature down converter in low-IF or direct-conversion receivers.
Yet another configuration of a quadrature converter is shown in FIG. 4. FIG. 4 shows quadrature up-converter 402. Quadrature up-converter 402 includes two mixers 404, 406, an oscillator 408 and an adder 410. For the configuration of the quadrature up-converter shown in FIG. 4, x is set to I and y is set to Q and the Q-channel output is either added to or subtracted from the I-channel output. In the case of quadrature up-converter 402, the output is real and has the form I cos ωtxt±Q sin ωtxt. Quadrature up-converter 402 functions as a quadrature modulator that modulates the baseband I and Q signals to a carrier frequency of ωtx. Such a quadrature modulator is frequently used to generate digital modulations, such as phase shift keying (PSK), quadrature amplitude modulation (QAM), frequency shift keying (FSK), orthogonal frequency division multiplexing (OFDM), and spread spectrum.
Many other configurations of quadrature converters exist. For instance, as shown in FIG. 5, two quadrature converters may be cross-coupled to form a converter 502. Converter 502 includes four mixers 504, 506, 508 and 510, an oscillator 512 and two adders 514, 516. Both the input and the output of converter 502 are complex signals. Thus, with the converter 502, true single sideband processing may be performed because the negative image of both the input as well as the output may be eliminated.
In practice, it is not possible to generate a perfect complex sinusoid using analog circuits due to process variation and asymmetry in layouts. In general, the I and Q channels will have different amplitudes and phases whereby the output of the quadrature converter will have xa cos(ωot+α)+jykb sin(ωot+β), where α is the gain of the I-channel, b is the gain of the Q-channel, α is the phase of the I-channel, and β is the phase of the Q-channel. In the case of a single sideband mixer configuration as shown in FIG. 2, the output may be represented by
                                                                        (                                                      a                    ⁢                                                                                  ⁢                                          ⅇ                                              -                        jα                                                                              +                                      b                    ⁢                                                                                  ⁢                                          ⅇ                                                                        -                          j                                                ⁢                                                                                                  ⁢                        β                                                                                            )                            ⁢                              u                ⁡                                  (                  t                  )                                                      2                    ⁢                      ⅇ                                          -                j                            ⁢                                                          ⁢                              ω                0                            ⁢              t                                      +                                                            (                                                      a                    ⁢                                                                                  ⁢                                          ⅇ                                              j                        ⁢                                                                                                  ⁢                        α                                                                              -                                      b                    ⁢                                                                                  ⁢                                          ⅇ                                              j                        ⁢                                                                                                  ⁢                        β                                                                                            )                            ⁢                              u                ⁡                                  (                  t                  )                                                      2                    ⁢                      ⅇ                                          +                                  jω                  0                                            ⁢              t                                                          Equation        ⁢                                  ⁢                  (          1          )                    
where u(t) is the input to the mixer. It is apparent that an unwanted signal
                    (                              a            ⁢                                                  ⁢                          ⅇ                              j                ⁢                                                                  ⁢                α                                              -                      b            ⁢                                                  ⁢                          ⅇ                              j                ⁢                                                                  ⁢                β                                                    )            ⁢              u        ⁡                  (          t          )                      2    ⁢      ⅇ                  +                  jω          0                    ⁢      t      has been generated. In a super-heterodyne receiver, this unwanted signal corresponds to signal contents at −ωi and −2ωo+ωi as shown in FIG. 3B. In a direct-conversion receiver, this unwanted signal corresponds to the negative sidebands of the received baseband signal r(t). In both cases, the desired signal has been distorted. The amount of desired signal relative to the distortion is referred to as the image-reject ratio and may be expressed by
                    20        ⁢                                  ⁢                              log            10                    ⁡                      (                                          1                +                                  γ                  2                                +                                  2                  ⁢                  γ                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                      ϕ                      )                                                                                                  1                +                                  γ                  2                                -                                  2                  ⁢                                      γcos                    ⁡                                          (                      ϕ                      )                                                                                            )                                              Equation        ⁢                                  ⁢                  (          2          )                    
where γ=b/a is the amplitude mismatch and φ=β−α is the phase mismatch. These two parameters constitute the I-Q imbalance in the receiver.
At the transmitter, a quadrature modulator is typically used. With imbalanced phase and amplitude, the output of the modulator, assuming subtraction at the output, will have aI cos(ωtxt+α)−bQ sin(ωtx+β), which may be represented by
                                                        (                                                a                  ⁢                                                                          ⁢                  I                  ⁢                                                                          ⁢                                      ⅇ                                          j                      ⁢                                                                                          ⁢                      α                                                                      +                                  jbQ                  ⁢                                                                          ⁢                                      ⅇ                                          j                      ⁢                                                                                          ⁢                      β                                                                                  )                        2                    ⁢                      ⅇ                                          jω                α                            ⁢              t                                      +                                            (                                                a                  ⁢                                                                          ⁢                  I                  ⁢                                                                          ⁢                                      ⅇ                                                                  -                        j                                            ⁢                                                                                          ⁢                      α                                                                      -                                  jbQ                  ⁢                                                                          ⁢                                      ⅇ                                                                  -                        j                                            ⁢                                                                                          ⁢                      β                                                                                  )                        2                    ⁢                                    ⅇ                                                -                                      jω                    α                                                  ⁢                t                                      .                                              Equation        ⁢                                  ⁢                  (          3          )                    
The distortion due to the imbalance is clear in Equation 3 because the ideal transmitted baseband signal is I+jQ while the actual baseband signal has become aIejα+jbQejβ. Similarly, the parameters γ=b/a and φ=β−α constitute the I-Q imbalance in the transmitter.
To distinguish between the imbalance parameters at the receiver and transmitter, parameters associated with the receiver will have a subscript r and those associated with the transmitter will have a subscript t. For instance, γr=br/ar and φr=βr−αr denote the gain and phase mismatches at the receiver whereas γt=bt/αt and φt=βt−αt denote the gain and phase mismatches at the transmitter. Ignoring the higher order frequency terms, the following general model may be derived for the signal at the receiver output given both transmitter and receiver imbalance
                                                                                                                                    G                      c                                        ⁢                                          a                      r                                        ⁢                                          a                      t                                        ⁢                                                                  ⅇ                                                  j                          ⁡                                                      (                                                                                          α                                t                                                            -                                                              α                                r                                                                                      )                                                                                              ⁡                                              (                                                  1                          +                                                                                    γ                              r                                                        ⁢                                                          ⅇ                                                                                                -                                  j                                                                ⁢                                                                                                                                  ⁢                                                                  ϕ                                  r                                                                                                                                                                    )                                                              ⁢                                          (                                              I                        +                                                                              jγ                            t                                                    ⁢                          Q                          ⁢                                                                                                          ⁢                                                      ⅇ                                                          j                                                              ϕ                                t                                                                                                                                                        )                                                        4                                ⁢                                  ⅇ                                      j                    ⁡                                          (                                                                                                    ω                            IF                                                    ⁢                          t                                                +                                                  Δ                          ⁢                                                                                                          ⁢                          ω                          ⁢                                                                                                          ⁢                          t                                                +                        Δθ                                            )                                                                                  +                                                                                                                                G                    c                                    ⁢                                      a                    r                                    ⁢                                      a                    t                                    ⁢                                                            ⅇ                                              -                                                  j                          ⁡                                                      (                                                                                          α                                t                                                            -                                                              α                                r                                                                                      )                                                                                                                ⁡                                          (                                              1                        -                                                                              γ                            r                                                    ⁢                                                      ⅇ                                                          j                              ⁢                                                                                                                          ⁢                                                              ϕ                                r                                                                                                                                                        )                                                        ⁢                                      (                                          I                      -                                              j                        ⁢                                                                                                  ⁢                                                  γ                          t                                                ⁢                        Q                        ⁢                                                                                                  ⁢                                                  ⅇ                                                      -                                                          j                                                              ϕ                                t                                                                                                                                                                          )                                                  4                            ⁢                              ⅇ                                  -                                      j                    ⁡                                          (                                                                                                    ω                            IF                                                    ⁢                          t                                                +                                                  Δ                          ⁢                                                                                                          ⁢                          ω                          ⁢                                                                                                          ⁢                          t                                                +                        Δθ                                            )                                                                                                                              Equation        ⁢                                  ⁢                  (          4          )                    
where Gc is the gain of the transmission channel, ωIF=ωtx−ωo is the IF frequency, Δω is the frequency offset, and Δθ is the phase offset. This model is used extensively to determine the appropriate compensation factors in the receiver and transmitter to reduce the distortions due to I-Q imbalance.
I-Q imbalances introduce distortions in the transmitter and receiver. Imbalances are the result of asymmetry in circuit layouts and non-uniformity in IC fabrication processes (such as threshold mismatch and device mismatches). Typical RF transceivers operating at a few GHz may achieve 2 degrees and 2% of phase and amplitude mismatches even with careful layouts. While the mismatches seem small, they introduce additional distortions in RF systems so that bit-error rate is increased. Also, the mismatches worsen for higher carrier frequencies, for example millimeter wave. I-Q imbalance is especially detrimental to high-performance RF systems that use high-order modulations. Such high-performance RF systems include wireless local area networks such as IEEE 802.11a, broadband personal area networks such as IEEE 802.15.3, fixed wireless access such as Local Multipoint Distribution System (LMDS) and IEEE 802.16, and 2.5G/3G cellular systems.